Solve each equation using the Quadratic Formula.

4x²+x=1 .

Answer:

1.

\(C.\\sin(N)=\frac{\sqrt{3} }{2}\)

2.

\(x=82.1\)

3.

x = 18°

Step-by-step explanation:

1. The sine ratio is sin(θ) = opposite/hypotenuse, where θ is the reference angle.  When N is the reference angle, we see that side OP with a measure of 5√3 units is the opposite side and side NP with a measure of 10 units is the hypotenuse.

Thus, we can find plug everything into the sine ratio and simplify:

\(sin(N)=\frac{5\sqrt{3} }{10} \\\\sin(N)=\frac{\sqrt{3} }{2}\)

2. We can use the tangent ratio to solve for x, which is tan (θ) = opposite/adjacent.  If we allow the 75° to be our reference angle, we see that the side measuring x units is the opposite side and the side measuring 22 units is the adjacent side.  Thus, we can plug everything into the ratio and solve for x or the measure of the opposite side:

\(tan(75)=\frac{x}{22}\\ \\22*tan(75)=x\\\\82.10511777=x\\\\82.1=x\)

3. Since we're now solving for an angle, we must using inverse trigonometry.  We can use the inverse of the tangent ratio, whose equation is tan^-1 (opposite/adjacent) = θ.  We see that when the x° is the reference angle, the side measuring 11 units is the opposite and the side measuring 33 units is the adjacent side.  Now we can do the inverse trig to find the measure of x:

\(tan^-^1(\frac{11}{33})=x\\ 18.43494882=x\\18=x\)

297 children like at most one kind of ice cream.

Total number of children = 397

Let C and P be the number of children like chocolate and pistachio ice cream respectively.

n(C) = 175

n(P) = 225

The number of student do not like chocolate or pistachio ice cream is,

n(P' and C') = 97

n(P or C)' = 97, by De Moivre's theorem

n(P or C) = 397-97 = 300

n(P) + n(C) - n(P and C) = 300

175 + 225 - n(P and C) = 300

400 - n(P and C) = 300

n(P and C) = 100

So the required number of children like at most one kind of ice cream = 397-n(P and C) = 397-100 = 297

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 The sample we collect plays a cruciThe sample we collect can significantly affect our understanding of a population,

especially when it comes to making generalizations or drawing conclusions about the entire population based on the sample data. The key aspect in this regard is whether the sample is representative of the population.

A representative sample is one that accurately reflects the characteristics, diversity, and distribution of the population from which it is drawn. When our sample is representative, we can have greater confidence in generalizing the findings from the sample to the larger population. However, if our sample is not representative, our understanding of the population may be biased or limited.

In the provided example, the given data represents different age groups and their reported time spent with friends per day. If we were to collect a sample from this population, it would be crucial to ensure that the sample includes individuals from each age group in proportion to their representation in the population. This would help ensure that our sample accurately reflects the age distribution of the population.

If our sample is not representative, it may lead to inaccurate conclusions. For instance, if we only surveyed individuals from the age group 55-64 and ignored the other age groups, our understanding of the population's time spent with friends would be limited to that specific age group. We wouldn't be able to generalize our findings to the entire population because we would have missed important variations in behavior across other age groups.

To improve the representativeness of our sample, we could use random sampling techniques, such as simple random sampling or stratified sampling, to ensure that individuals from all age groups are included in the sample. By doing so, we can enhance our understanding of the population as a whole and make more accurate inferences.

In conclusion, the sample we collect plays a crucial role in shaping our understanding of a population. A representative sample enables us to make valid generalizations and draw reliable conclusions about the entire population. It ensures that the characteristics and diversity of the population are properly reflected in the sample, allowing for more accurate insights and informed decision-making.al role in shaping our understanding of a population

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-18 is the y intercept

The slope is in y=mx+b form,

And b is the y intercept. Mx is the slope(m) and x is the variable, in this case 8x

So slope is 8, and y intercept is -18

Answer: x = 3.5

Step-by-Step Solution:

Let us first label the figure.

Let the Triangle be ABC with a line DE || BC.

Now, in ∆ABC,

DE || BC (given)

=> AD/DB = AE/EC (by B.P.T)

Substituting the given values,

AD/DB = AE/EC

2/4 = x/7

1/2 = x/7

2x = 7

x = 7/2

=> x = 3.5

Therefore, x = 3.5

Answer:

3000meter

Step-by-step explanation:

\( \frac{300000}{100} = 3000\)

the chance that the weight of a product is between 115 LB and 118 LB is approximately 0.1207, or 12.07%.

To find the probability that the weight of a product is between 115 LB and 118 LB, we can use the properties of the normal distribution.

Given that the weight of products follows a normal distribution with a mean (µ) of 120 LB and a standard deviation (σ) of 30 LB, we can standardize the values of 115 LB and 118 LB using the z-score formula:

z = (x - µ) / σ

For 115 LB:

z1 = (115 - 120) / 30 = -0.1667

For 118 LB:

z2 = (118 - 120) / 30 = -0.0667

Next, we can use a standard normal distribution table or a calculator to find the corresponding probabilities for these z-values.

Using a standard normal distribution table, the probability of having a z-value between -0.1667 and -0.0667 is approximately 0.1207.

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Answer:

x = 16

Step-by-step explanation:

The two angles are vertical angles

Vertical angles are congruent

That being said we can create an equation

4x - 7 = 3x + 9

now we solve for x

step 1 add 7 to each side

9 + 7 = 16

-7 + 7 cancels out

now we have 4x = 3x + 16

step 2 subtract 3x from each side

4x - 3x = x

3x - 3x cancels out

we're left with x = 16

Answer:

i think it's like this

1/2-2/5

5-4/10

1/10

The required bearing angle of town B from Thomas is 75°.

We have,

Bearing is basically an angle that is measured clockwise from the north. Bearing are generally written in three figure.

Given that

Thomas is facing towards town A, which is at a bearing of 300°.

Implies that town A is 300° from north.

If Thomas turns 135° clockwise, then he faces towards town B,

The bearing angle will be 300+135 = 435°

Since, one complete round makes angle 360°, therefore

The required bearing angle = 435 - 360 = 75

The bearing angle of town B from Thomas is 75°.

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Answer:

Chad's other friend pick 2 pounds of blue berries.

Step-by-step explanation:

Chad picked 2 pounds of blue berries.

One of his friends picked 1 pound of blue berries.

Totally there is 5 pounds.

So, other friend picked blue berries = 5-2-1 =2 pounds

We get answer as whole number. So no need to write it in mixed form.

\(a {y}^{2} + by + c = 0 \)

\(3 {y}^{2} + 8y + 2 = 0 \)

\(a = 3\)

\(b = 8\)

\(c = 2\)

_________________________________

\(∆ = {b}^{2} - 4ac \)

\(∆ = ({8})^{2} - 4 \times (3) \times (2) \)

\(∆ = 64 - 24\)

\(∆ = 40\)

_________________________________

\(y = \frac{ -b ± \sqrt{∆} }{2a} \\ \)

##############################

\(y(1) = \frac{ - 8 + \sqrt{40} }{6} \\ \)

\(y(1) = \frac{ - 8 + \sqrt{4 \times 10} }{6} \\ \)

\(y(1) = \frac{ - 8 + 2 \sqrt{10} }{6} \\ \)

\(y(1) = \frac{2( - 4 + \sqrt{10}) }{2 \times 3} \\ \)

\(y(1) = \frac{ - 4 + \sqrt{10} }{3} \\ \)

+++++++++++++++++++++++++++++++++++++++

\(y(2) = \frac{ - 8 - \sqrt{40} }{6} \\ \)

\(y(2) = \frac{ - 8 - \sqrt{4 \times 10} }{6} \\ \)

\(y(2) = \frac{ - 8 - 2 \sqrt{10} }{6} \\ \)

\(y(2) = \frac{2( - 4 - \sqrt{10}) }{2 \times 3} \\ \)

\(y(2) = \frac{ - 4 - \sqrt{10} }{3} \\ \)

##############################

_________________________________

And we're done.....♥️♥️♥️♥️♥️

The value of positive integers in the set are 16.

According to the statement

we have given that the there is a set of numbers from 1 to n and we have to find that the how many integers in this set. and there is one condition that the numbers in the set are less than or equal to 24.

So, For this purpose,

Since \($1 + 2 + \cdots + n = \frac{n(n+1)}{2}$\)

the condition is equivalent to having an integer value for \($\frac{n!} {\frac{n(n+1)}{2}}$.\)

This reduces, when \($n\ge 1$\), to having an integer value for \($\frac{2(n-1)!}{n+1}$\)

This fraction is an integer unless n+1 is an odd prime. There are 8 odd primes less than or equal to 24,

so there are 24-8 = 16.

So, The value of positive integers in the set are 16.

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The conformal map from the horizontal strip {-a < Im(z) < a} onto the right half-plane {Re(w) > 0} is given by h(z) = \(e^(πiz / a)\).

What is exponential function?

The exponential function is a function of the form f(x) = \(e^x\), where e is Euler's number (approximately equal to 2.71828) and x is the input variable. The exponential function is commonly used in various areas of mathematics, physics, and engineering due to its fundamental properties.

The exponential function can be used to locate a conformal projection onto the right half-plane Re(w) > 0 from the horizontal strip -a Im(z) a. onto the right half-plane {Re(w) > 0}, we can use the exponential function. The key is to map the strip onto the upper half-plane first, and then apply another transformation to map the upper half-plane onto the right half-plane.

Step 1: Map the strip onto the upper half-plane

Consider the function f(z) = \(e^(πiz / (2a)\)). This function maps the strip {-a < Im(z) < a} onto the upper half-plane.

Step 2: Map the upper half-plane onto the right half-plane

To map the upper half-plane onto the right half-plane, we can use the transformation g(w) = w², which squares the complex number w.

Combining these two steps, we have the conformal map from the horizontal strip onto the right half-plane:

h(z) = g(f(z)) = \((e^(πiz / (2a))\))² = \(e^(πiz / a)\).

Therefore, the conformal map from the horizontal strip {-a < Im(z) < a} onto the right half-plane {Re(w) > 0} is given by h(z) = \(e^(πiz / a)\).

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Nash equilibrium describes a situation in which no firm can improve its outcome by independently changing its course of action.

In the context of oligopoly, a Nash equilibrium refers to a situation in which no firm can improve its outcome by unilaterally changing its course of action, given the actions of other firms.

In other words, each firm is choosing its best response to the actions of other firms, and no firm has an incentive to deviate from its chosen strategy.

Nash equilibrium is a concept derived from game theory, which is the study of strategic decision-making in competitive situations. It helps to predict the behavior and outcomes in situations where multiple players or firms are interdependent.

Hence the correct option is Nash equilibrium.

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The expression representing the situation is B = 3M + d, where B represents the number of baseball cards Bobby has, M represents the number of baseball cards Michael has, and d represents the additional amount that Bobby has compared to three times the number of cards Michael has.

Step 1: Assign variables.

Let's assign the variable "B" to represent the number of baseball cards Bobby has and the variable "M" to represent the number of baseball cards Michael has.

Step 2: Understand the relationship.

According to the given information, Bobby has "d" more than 3 times the number of baseball cards as Michael. This means that Bobby's number of baseball cards can be calculated by taking 3 times the number of cards Michael has and adding "d" to it.

Step 3: Create the expression.

To represent the situation, we can write the expression as: B = 3M + d.

Step 4: Interpret the expression.

In this expression, "3M" represents 3 times the number of baseball cards Michael has, and "d" represents the additional amount that Bobby has compared to that.

Therefore, the expression B = 3M + d represents the situation where Bobby has "d" more than 3 times the number of baseball cards as Michael. This expression allows us to calculate Bobby's number of cards based on the given relationship between their card counts.

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Answer:

$12.50

Step-by-step explanation:

62% would be left,so

\(\frac{62}{100}\) × $20

=$12.50

The given information is as follows: R(x) = 4x and C(x) = 0.001x² +1.7x + 50 We need to find the following:
P(100)

Find the production level that results in the maximum profit
P(100):To find P(100), we first need to find P(x) since we are given R(x) and C(x). We know that P(x) = R(x) - C(x). Hence:
P(x) = 4x - (0.001x² +1.7x + 50)
P(x) = 4x - 0.001x² -1.7x - 50
P(x) = - 0.001x² + 2.3x - 50
Now, we can find P(100) by substituting x = 100 into the expression for P(x).
P(100) = -0.001(100)² + 2.3(100) - 50
P(100) = -0.001(10000) + 230 - 50
P(100) = -10 + 230 - 50
P(100) = 170

The profit from the production and sale of 100 items is $170. Find the production level that results in the maximum profit:To find the production level that results in the maximum profit, we need to find the value of x that maximizes P(x). Let P′(x) be the derivative of P(x).
P(x) = - 0.001x² + 2.3x - 50
P′(x) = - 0.002x + 2.3
We can find the critical points of P(x) by solving P′(x) = 0.
- 0.002x + 2.3 = 0
x = 1150
We can use the second derivative test to determine whether this critical point results in a maximum profit.
P′′(x) = -0.002 (which is negative)Since P′′(1150) < 0, the critical point x = 1150 corresponds to a maximum profit. Given R(x) = 4x and C(x) = 0.001x² +1.7x + 50, we are required to determine the profit function P(x) and find the profit for the production of 100 items, as well as the level of production that results in the maximum profit.Using the information that the revenue R(x) equals 4x, we can write the profit P(x) function as P(x) = R(x) - C(x) = 4x - (0.001x² +1.7x + 50) = - 0.001x² + 2.3x - 50. This gives us the profit for the production of any number of items.We can then find the profit for the production of 100 items by substituting x = 100 in the P(x) function: P(100) = -0.001(100)² + 2.3(100) - 50 = 170.This means that the profit from the production and sale of 100 items is $170.To find the level of production that results in the maximum profit, we can find the critical points of P(x) by solving P′(x) = -0.002x + 2.3 = 0. This gives us the critical point x = 1150. We can then use the second derivative test to check if x = 1150 is a maximum. Since P′′(1150) < 0, we can conclude that the level of production that results in the maximum profit is 1150.

Therefore, the profit from the production and sale of 100 items is $170, and the level of production that results in the maximum profit is 1150 items.

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The independent set in a graph is a set of vertices that contain no edges. So, no neighboring vertices are included. The max independent set problem is to get an independent set of maximum size in graph G.

The solution for this question is discussed below:

a) The integer linear program for the max independent set problem is as follows:

maximize ∑x_i Subject to: x_i+x_j ≤ 1 {i,j} ∈ E;x_i ∈ {0, 1} ∀i. The variable x_i can represent whether the ith vertex is in the independent set. It can take on two values, either 0 or 1.

b) The LP relaxation for the max independent set problem is as follows:

Maximize ∑x_iSubject to:

xi+xj ≤ 1 ∀ {i, j} ∈ E;xi ≥ 0 ∀i. The variable xi can take on fractional values in the LP relaxation.

c) The family of graphs is as follows:

Consider a family of graphs G = (V, E) defined as follows. The vertex set V has n = 2^k vertices, where k is a positive integer. The set of edges E is defined as {uv:u, v ∈ {0, 1}^k and u≠v and u, v differ in precisely one coordinate}. It can be shown that the size of the max independent set is n/2. Using LP, the value can be determined. LP provides a value of approximately n/4. Therefore, the ratio LP-OPT/OPT is at least c/4. Therefore, the ratio is in for a constant c>0.

d) The size of a max-independent set is equivalent to the number of vertices minus the minimum vertex cover size.

e) The 2-approximation algorithm for vertex cover implies a 2-approximation algorithm for the max independent set.

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The cost of the sheets is less for the second example, the second example is the better deal.

Consider the first example:

Here, A brand of sheets is on sale for “Buy one get one at 50% off”. If each sheet costs $18.00.

So, the value of the first sheet will be $18,

And since the other sheet is at 50% off .

That is it will be half of the original value

= 50/100 * 18

=9

Therefore, the cost of two sheets

=$18 + $9

= $27.

So, Six sheets will cost

=$27+$27+$27

= $81

Cost of six sheets will be $81

Now, consider the second example

In the second example sheets are 30% off.

So, 30% of 18

= $6.

Subtract that value from 18 and you will get $12.

So, cost of one sheet =  $12.

cost of 6 sheet

=  $12×6

= $72  

In the first example, Six sheets costs $81, while in the second example six sheets cost $72.

Since the cost is less for the second example. So, the second example is the better deal.

Answer:

The answer is 1825.35

Step-by-step explanation:

Since the first week isn’t included, we only count 4 weeks.

So first, 2000 x 0.03 = 60

(By the way 0.03 is 3%)

2000 - 60 = 1940

Subtracting 2000 by 60 represents the amount of money the aquarium went down in that week.

Then, week 3:

1940 x 0.03 = 58.2

1940 - 58.2 = 1881.8

Now, the last week, week four:

1881.8 x 0.03 = 56.45

1881.8 - 56.45 = 1825.35.

Therefore, 1825.35 after 5 weeks.

Answer:16

Step-by-step explanation:

The null hypothesis for this test is that the mean ECLSE test score for the university is equal to 1200. The alternative hypothesis is that the mean is not equal to 1200.

To perform the test, you will need to calculate the t-statistic and the p-value. The t-statistic is calculated by:

t = (sample mean - hypothesized mean) / (sample standard deviation/sqrt (sample size))

In this case, the sample mean is 1180, the hypothesized mean is 1200, the sample standard deviation is 100, and the sample size is 100. Plugging these values into the equation gives:

t = (1180 - 1200) / (100 / sqrt(100)) = -20 / 10 = -2

Next, you will need to calculate the p-value. The p-value is the probability of observing a t-statistic as extreme as the one calculated, given that the null hypothesis is true.

To calculate the p-value, you will need to use a t-distribution table or a software package to find the t-critical value for a one-tailed test with 99 degrees of freedom (since you have a sample size of 100) and a significance level of 0.05. The t-critical value is the value that defines the rejection region for the test. If the t-statistic falls in the rejection region, you can reject the null hypothesis.

If the p-value is less than the significance level of 0.05, you can reject the null hypothesis and conclude that the mean ECLSE test score for the university is not equal to 1200 at the 95% significance level. If the p-value is greater than or equal to 0.05, you cannot reject the null hypothesis and cannot conclude that the mean is different from 1200.

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Answer:

27.88 ounces

Step-by-step explanation:

6.8 x 4.1 = 27.88 ounces

The last term is 300

The sum of all the terms is 15150.0

The average of all the terms is 151.5

Here is an example solution in Python that follows the given pseudocode:

# Prompt for and read the first term

first_term = int(input("Enter first term: "))

# Prompt for and read the common difference

common_difference = int(input("Enter common difference: "))

# Prompt for and read the number of terms

number_of_terms = int(input("Enter number of terms: "))

# Calculate the last term

last_term = first_term + (number_of_terms - 1) * common_difference

# Calculate the sum of all the terms

sum_of_terms = number_of_terms * (first_term + last_term) / 2

# Calculate the average of all the terms

average_of_terms = sum_of_terms / number_of_terms

# Display the results

print("The last term is", last_term)

print("The sum of all the terms is", sum_of_terms)

print("The average of all the terms is", average_of_terms)

If you run this code and enter the values from the sample run (first term: 3, common difference: 3, number of terms: 100), it will produce the following output:

The last term is 300

The sum of all the terms is 15150.0

The average of all the terms is 151.5

The program prompts the user for the first term, common difference, and number of terms. Then it calculates the last term using the given formula. Next, it calculates the sum of all the terms and the average of all the terms using the provided formulas. Finally, it displays the calculated results.

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The value of measure of PA,

PA = 4.62 units

Given that;

Parallelogram AOBR is shown.

Hence, By Pythagoras theorem,

We get;

OR² = 12² + 5²

OR² = 144 + 25

OR² = 169

OR = 13

Hence, By using proportionality theorem we get;

OR / AR = OA / PA

13 / 12 = 5 / PA

PA = 60 / 13

PA = 4.62 units

Thus, The value of measure of PA,

PA = 4.62 units

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Answer:

1 hour= 45 miles

1/2 hour=22.5 miles

3 hours= 135

Step-by-step explanation:

just work out one hour by halving 90 miles which is 45. Then work out the rest by halving again and multiplying by 3.

Answer:

1st way:

90/2=45

1 hour = 45 miles

45/2=22.5

0.5 hour = 22.5 miles

45+22.5=67.5

1.5 hour = 67.5 miles

67.5x2=135

3 hours = 135 miles

2nd way:

90 miles - 2 hours

1 hour - 60 minutes

60x2=120

90/120=0.75

1 minute - 0.75 miles

0.75x60=45

1 hour = 45 miles

60:2=30

60+30=90

0.75x90=67.5

1.5 hour = 67.5 miles

60x3=180

0.75x180=135

3 hours = 135 miles